Topological Methods in Nonlinear Analysis

Realization of a graph as the Reeb graph of a Morse function on a manifold

Łukasz Patryk Michalak

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 749-762.

Dates
First available in Project Euclid: 9 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1533780029

Digital Object Identifier
doi:10.12775/TMNA.2018.029

Mathematical Reviews number (MathSciNet)
MR3915662

Zentralblatt MATH identifier
07051691

Citation

Michalak, Łukasz Patryk. Realization of a graph as the Reeb graph of a Morse function on a manifold. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749--762. doi:10.12775/TMNA.2018.029. https://projecteuclid.org/euclid.tmna/1533780029


Export citation

References

  • S. Biasotti, D. Giorgi, M. Spagnuolo and B. Falcidieno, Reeb graphs for shape analysis and applications, Theoret. Comput. Sci. 392 (2008), 5–22.
  • K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan and V. Pascucci, Loops in Reeb graphs of $2$-manifolds, Discrete Comput. Geom. 32 (2004), 231–244.
  • I. Gelbukh, Loops in Reeb graphs of $n$-manifolds, Discrete Comput. Geom. 59 (2018), no. 4, 843–863.
  • I. Gelbukh, The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations, Math. Slovaca 67 (2017), 645–656.
  • M. Kaluba, W. Marzantowicz and N. Silva, On representation of the Reeb graph as a sub-complex of manifold, Topol. Methods Nonlinear Anal. 45 (2015), no. 1, 287–307.
  • M.A. Kervaire and J.W. Milnor, Groups of Homotopy Spheres I, Ann. of Math. (2), vol. 77, no. 3. (May, 1963), 504–537.
  • J. Martinez-Alfaro, I.S. Meza-Sarmiento and R. Oliveira, Topological classification of simple Morse Bott functions on surfaces, Contemp. Math. 675 (2016), 165–179.
  • Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math. 65 (2011), no. 1, 75–84.
  • J.W. Milnor, Differential Topology, Lectures on Modern Mathematics, Vol. II, Wiley, New York, 1964, 165–183.
  • J.W. Milnor, Lectures on the h-Cobordism Theorem, Princeton University Press, Princeton, 1965.
  • G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, C.R. Acad. Sci. Paris 222 (1946), 847–849.
  • V.V. Sharko, About Kronrod–Reeb graph of a function on a manifold, Methods Funct. Anal. Topology 12 (2006), 389–396.
  • F. Takens, The minimal number of critical points of a function on a compact manifold and the Lusternik–Schnirelman category, Invent. Math. 6 (1968), 197–244.